Geodesic
One sufficient condition for Hamiltonian graphs involving distances
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2012), pp. 46-52

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Let $G$ be a 2-connected graph of order $n$ such that $2|N(x)\cup N(y)|+d(x)+d(y)\geq2n-1$ for each pair of nonadjacent vertices $x,y$. Then, as was proved in 1990 by G. T. Chen, $G$ is Hamiltonian. In this paper we introduce one more condition and prove that if $G$ is a 2-connected graph of order $n$ and $2|N(x)\cup N(y)|+d(x)+d(y)\geq2n-1$ for each pair of nonadjacent vertices $x,y$ such that $d(x,y)=2$, then $G$ is Hamiltonian.
Keywords: Hamiltonian graph, neighborhood union condition, new sufficient condition.
Mots-clés : Ore condition, Chen condition 
@article{IVM_2012_4_a4,
     author = {Kewen Zhao and Lin Yue and Zhang Ping},
     title = {One sufficient condition for {Hamiltonian} graphs involving distances},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {46--52},
     publisher = {mathdoc},
     number = {4},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2012_4_a4/}
}
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Kewen Zhao; Lin Yue; Zhang Ping. One sufficient condition for Hamiltonian graphs involving distances. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2012), pp. 46-52. http://geodesic.mathdoc.fr/item/IVM_2012_4_a4/